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I'm trying to map integer $x$ to the set of all (ordered) $n$-tuples of positive integers that sum to it.

$ f_n(x): \Bbb{Z} \to \cal{P}\left({\Bbb{Z}^{+}}^{n}\right) = \left\{\left(i_1,\dots,i_n\right)\right |\sum{i_k}=x\}$

Is there a name for this (admittedly partial) function?

Example:

$ (3, 5) \mapsto \left\{\left(1, 1, 3\right), \left(1, 2, 2\right), \left(1, 3, 1\right), \left(2, 1, 2\right), \left(2, 2, 1\right), \left(3, 1, 1\right)\right\} $

JamesTheAwesomeDude
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  • One might call them ordered partitions of $x$ of length $n$. For more on partitions, see https://en.wikipedia.org/wiki/Partition_(number_theory) – morrowmh Jul 22 '22 at 20:16
  • This is the $n$-dimensional sphere of radius $x$ in $\mathbb Z^{+n}$, with the taxicab metric – Andrei Jul 22 '22 at 20:37

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